Acta Mathematica

Volume 219 (2017)

Number 1

Singular Ricci flows I

Pages: 65 – 134

DOI: https://dx.doi.org/10.4310/ACTA.2017.v219.n1.a4

Authors

Bruce Kleiner (Courant Institute of Mathematical Sciences, New York, N.Y., U.S.A.)

John Lott (Department of Mathematics, University of California at Berkeley)

Abstract

We introduce singular Ricci flows, which are Ricci flow spacetimes subject to certain asymptotic conditions. These provide a solution to the long-standing problem of finding a good notion of Ricci flow through singularities, in the $3$-dimensional case.

We prove that Ricci flow with surgery, starting from a fixed initial condition, subconverges to a singular Ricci flow as the surgery parameter tends to zero. We establish a number of geometric and analytical properties of singular Ricci flows.

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Research supported by NSF grants DMS-1105656, DMS-1207654 and DMS-1405899, and a Simons Fellowship.

Received 16 November 2015

Accepted 28 October 2017

Published 31 January 2018